Using the Lagrange mean-value theorem, the author proves that \[ (1)\quad \int^{b}_{a}f(t)dt\geq (b-a)f(\frac{a+b}{2}), \]
\[ (2)\quad 2\int^{b}_{a}f(t)dt\leq (b-a)f(\sqrt{ab})+(\sqrt{b}- \sqrt{a})(\sqrt{b}f(b)-\sqrt{a}f(a))\quad (in\quad (2)\quad 0\leq a<b) \] provided f: [a,b]\(\to {\mathbb{R}}\) is a differentiable function with increasing derivative on [a,b] (Theorem 1). The inequality (1) is generalized in Theorem 2 and the paper is concluded with some interesting applications for the logarithmic and exponential functions.